A non-local approach to the generalized Stokes operator with bounded measurable coefficients

Abstract

We establish functional analytic properties of the Stokes operator with bounded measurable coefficients on $L_{\sigma }^p({\mathbb{R}}^d)$, $d\ge 2$, for $|1/p-1/2|<1/d$. These include optimal resolvent bounds and the property of maximal $L^q$-regularity. We further give regularity estimates on the gradient of the solution to the Stokes resolvent problem with bounded measurable coefficients. As a key to these results we establish the validity of a non-local Caccioppoli inequality to solutions of the Stokes resolvent problem.